Fourier transforms and convolutions

This chapter revisits the essential mathematics of integral calculus and introduces the important concept of the Fourier transform, the properties of which are elegantly demonstrated by sketching the curves of various functions. It demonstrates the essential principles of diffraction by determining the Fourier transforms of regularly repeating patterns which can be represented mathematically by Dirac delta functions — the very important concept of Fourier space (or reciprocal space) follows from this discussion. This section leads into a description of another highly important mathematical concept, the convolution. Convolutions allow two functions to be combined and provide an extremely elegant mathematical description of the crystalline state as well as an insight into one of crystallography's most important structure-solving tools, the Patterson function.